Studying near-field propagation of generalized parabolic beams by virtual source method

Abstract

Generalized parabolic beams have various optical morphologies. They can be used in different research fields, such as component design, aero-optics, and microwave wireless power transmission. Studying the near-field transmission characteristics of these beams is important for improved utilization. We develop a more accurate theoretical framework to precisely understand the propagation behavior of complex light fields in the near-field range, especially to overcome the limitations of conventional near-axis approximation. This framework fully reveals the propagation mechanism of parabolic beams and their energy transmission modes. Here, based on the principle of independent propagation and the virtual source method, a group of virtual sources is introduced to analyze generalized parabolic beams. These beams can be expanded into the superposition of infinite continuous integer Bessel beams. Then, by combining the Weber integral formula and the Fourier Bessel transform, we rigorously derive an integral expression for generalized parabolic beams during near-field propagation. This expression breaks the limits of the traditional paraxial approximation and contains all the key propagation parameters of the family of beams. Based on this integral expression, the intensity distribution and phase characteristics of the generalized parabolic beam along the optical axis are further calculated and analyzed to reveal its energy transfer mode and phase characteristics. By comparing the paraxial approximate solution and the nonparaxial corrected solution for generalized parabolic beams, the far-field propagation of generalized parabolic beams is found to be the same when the propagation distance is sufficiently large. Such simulation results indirectly confirm the correctness of the obtained theoretical solution. The simple paraxial approximation theory can conveniently calculate the far-field propagation of generalized parabolic beams. However, large errors exist when paraxial theory is used to calculate the near-field distribution of generalized parabolic beams. Although calculating nonparaxial propagation is especially complex, the nonparaxial correction solution is necessary when generalized parabolic beams are used in near-field research. Such research results not only deepen the understanding of the propagation mechanism of generalized parabolic beams but also lay a theoretical foundation for the calculation of precise propagation behaviors of other complex light fields in near-field optics.